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arXiv · 2605.15107

Solutions for Hecke Sum Questions of Banerjee and Bringmann

George E. Andrews·Mohamed El Bachraoui
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Abstract

The present authors introduced a two-color partition series $S(q)$ and conjectured a Hecke-type formula for the even part of $(q^4;q^4)_\infty S(q)$. Banerjee and Bringmann proved the conjecture by using indefinite theta functions, modular completions, and Sturm's theorem. They also asked whether a direct proof, for instance one based on Bailey-type ideas, could be found, and they suggested that the odd residue classes may be worth studying. We prove a two-variable refinement with an additional parameter $a$. Our proof relies entirely on $q$-series combined with the Bailey pairs The original even identity and the odd identity then follow as corollaries by letting $a=1$. We also record parameter symmetries and cyclotomic companions, including a vanishing result at $a=i$.

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George E. Andrews, Mohamed El Bachraoui. 2026-05-14. Solutions for Hecke Sum Questions of Banerjee and Bringmann. https://arxiv.org/abs/2605.15107

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